The statement R=C=1 is nonsensical.Suppose R = C = 1 then the transfer function from the input voltage to the voltage across the capacitor is 1/ (s+1). So Vc(S) / V(S) = 1/(s+1). Getting back to time-domain: Vc(t) = V(t) * e^-t. What's wrong here? Thank you!
If C is capacitance measured in farads, then it is a real number. The reactance of C, Xc, with units of ohms is also a real number. The impedance or admitaance of a capacitor C is a complex number. In the ideal case it would have a zero real part. For a real component there will be some equivalent series resistance.C is complex, R isn't.